Begin with the Maclaurin (Taylor) series: cos r = 1 - 2! 4! 6! (a) Evaluate cos 0. (b) List the zeros of cos r. (c) Expand cos r as an infinite product in the form Pla) = (1 - ) (1 - #) (1 -). where a, b, c, ... are the zeros of cos r.
Begin with the Maclaurin (Taylor) series: cos r = 1 - 2! 4! 6! (a) Evaluate cos 0. (b) List the zeros of cos r. (c) Expand cos r as an infinite product in the form Pla) = (1 - ) (1 - #) (1 -). where a, b, c, ... are the zeros of cos r.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Begin with the Maclaurin (Taylor) series:
cos r = 1 –
2!
4!
6!
(a) Evaluate cos 0.
(b) List the zeros of cos r.
(c) Expand cos r as an infinite product in the form
P(2) = (1 - 4) (1 - #) (1 - ).--
where a, b, c,... are the zeros of cos r.
(d) Equate the coefficients of the quadratic term in the Taylor expansion of
cos r to the corresponding coefficient in the infinite product representa-
tion of cos r obtained in part (c).
(e) Use the result in part (d) above to derive the sum of the reciprocals of
the squares of the odd integers:
1
1
+
9.
1+
+
81
+
25
49
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Follow-up Question
Kindly answer parts d and e too . I need answer as soon as possible
Transcribed Image Text:Begin with the Maclaurin (Taylor) series:
x²
2-6
-
cos .r = 1.
2!
4! 6!
(a) Evaluate cos 0.
(b) List the zeros of cos r.
(c) Expand cos e as an infinite product in the form
P(x) = (1 - ) (¹ - ) (¹-2)...
where a, b, c, ... are the zeros of cos.z.
(d) Equate the coefficients of the quadratic term in the Taylor expansion of
cos r to the corresponding coefficient in the infinite product representa-
tion of cos z obtained in part (c).
(e) Use the result in part (d) above to derive the sum of the reciprocals of
the squares of the odd integers:
1
1
1+ +
+ +
9 25 49
+
앗 100
11
Solution
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